Integrate cos(log(x)) dx

There are several approaches to integrating this function, thus making this a valuable exercise for students needing to develop fluency with the various techniques of integration and practise dealing with integrals that do not necessarily fit the pattern of textbook examples.The first approach to integrating a non-trivial function such always be to perform a substitution. In this case, we could substitute using t = log(x), differentiating to give dt = (1/x) dx. Bringing the x up to the other side gives dx = x dt. We've now found an expression for dx which can be substituted back into the original integral, giving the integral of x cos(log(x)) dt. We've already said that log(x) = t which gives the integral of x cos(t) dt. By laws of logarithms, we observe that if log(x) = t then x = exp(t), which we also substitute giving the integral of exp(t)cos(t) dt. The fact that all our terms in the integral are in terms of our new variable, t, signifies that we have complete our substitution. This new integral in t is trivial and can be easily solved applying the integration by parts formula twice, giving the answer. Don't forget the arbitrary constant (+c)!Another approach to integrating this function involves using Euler's formula (exp(ix) = cos(x) + isin(x)) using log(x) instead of x to give exp(i log(x)) = x^i = cos(log(x)) + isin(log(x)). Integrating both sides with respect to x and then equating the real parts gives an efficient (and elegant) solution to the integral which requires much less work! Happy days! This is a great example of how thinking about the nature of a problem and bringing in identities used in other areas of mathematics can often make solving the problem so much easier.